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Zbl 0548.47017
Reade, J.B.
On the spectrum of the Cesaro operator. (English)
[J] Bull. Lond. Math. Soc. 17, 263-267 (1985). ISSN 0024-6093; ISSN 1469-2120/e

The Cesàro operator $Cx=y$ where $y\sb n=\frac{x\sb 1+x\sb 2+...+x\sb n}{n}$ is shown to have spectrum $\vert\lambda -{1\over2}\vert\le {1\over2}$ when acting on the space $c\sb 0$ of sequences convergent to zero. C is shown to have no eigenvalues, whilst its adjoint $C\sp*$ has eigenvalues $\vert\lambda -{1\over2}\vert <{1\over2}$ all simple. The methods are similar to those of Halmos et. al. dealing with C on $\ell\sp 2$ (reference given) except that a direct proof of the invertibility of C-$\lambda$ I for $\vert\lambda -{1\over2}\vert >{1\over2}$ is needed.

MSC 2000:: *47B37 Operators on sequence spaces, etc.
47A10 Spectrum and resolvent of linear operators
40G05 Traditional summability methods

Keywords: Cesàro operator; spectrum; eigenvalues

Cited in: Zbl 1096.47031 Zbl 0779.47030 Zbl 0691.40004 Zbl 0814.47035 Zbl 0601.47004

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